Quiz — 680797f29ebc0_Correction Série N° 25 Etude De Fonction-Img.pdf

Question 1 sur 10 20:00

[{"id":6121,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 2x² + 5x - 7 ?","option_a":"3x² - 4x + 5","option_b":"x² - 4x + 5","option_c":"3x² - 4x + 5x","option_d":"x³ - 2x² + 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de x³ est 3x², celle de -2x² est -4x, celle de 5x est 5, et la dérivée de -7 est 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3x² - 4x + 5\", \"b\": \"x² - 4x + 5\", \"c\": \"3x² - 4x + 5x\", \"d\": ","_debug_options_count":4},{"id":6122,"question":"La fonction f(x) = (x² + 1)\/x est-elle définie en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction n'est pas définie en x = 0 car le dénominateur s'annule.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":6123,"question":"Quelle est la limite de f(x) = (2x² + 3x - 1)\/(x² - 4) quand x tend vers l'infini ?","option_a":"2","option_b":"∞","option_c":"0","option_d":"1\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par x², on obtient (2 + 3\/x - 1\/x²)\/(1 - 4\/x²), dont la limite est 2\/1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"2\", \"b\": \"∞\", \"c\": \"0\", \"d\": \"1\/2\"}}","_debug_options_count":4},{"id":6124,"question":"La fonction f(x) = x² - 4x + 3 est-elle croissante sur l'intervalle [2 ; +∞[ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 2x - 4 est positive pour x ≥ 2, donc la fonction est croissante sur cet intervalle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":6125,"question":"Quelle est l'équation de l'asymptote horizontale de f(x) = (3x + 1)\/(x - 2) ?","option_a":"y = 3","option_b":"y = 0","option_c":"y = -2","option_d":"y = 1\/3","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par x, on obtient (3 + 1\/x)\/(1 - 2\/x), dont la limite est 3\/1 = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"y = 3\", \"b\": \"y = 0\", \"c\": \"y = -2\", \"d\": \"y = 1\/3\"}}","_debug_options_count":4},{"id":6126,"question":"La fonction f(x) = e^x est-elle toujours positive ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle e^x est toujours strictement positive pour tout x réel.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":6127,"question":"Quelle est la dérivée de la fonction f(x) = ln(x² + 1) ?","option_a":"2x\/(x² + 1)","option_b":"2x","option_c":"(x² + 1)\/2x","option_d":"1\/(x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de dérivation des fonctions composées, f'(x) = (2x)\/(x² + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"2x\/(x² + 1)\", \"b\": \"2x\", \"c\": \"(x² + 1)\/2x\", \"d\": \"1\/(x² + 1)\"","_debug_options_count":4},{"id":6128,"question":"La fonction f(x) = x³ - 3x + 2 admet-elle un extremum local en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 3x² - 3 s'annule en x = 1, et f''(x) = 6x \u003E 0 pour x = 1, donc c'est un minimum local.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":6129,"question":"Quelle est la limite de f(x) = √(x² + 1) - x quand x tend vers +∞ ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"-∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"En multipliant par l'expression conjuguée, on obtient (x² + 1 - x²)\/(√(x² + 1) + x) = 1\/(√(x² + 1) + x), dont la limite est 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"∞\", \"d\": \"-∞\"}}","_debug_options_count":4},{"id":6130,"question":"La fonction f(x) = sin(x)\/x est-elle continue en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction n'est pas définie en x = 0, mais on peut prolonger par continuité en posant f(0) = 1 (limite quand x tend vers 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]